Jet schemes and invariant theory

Linshaw, Andrew R.; Schwarz, Gerald W.; Song, Bailin

doi:10.5802/aif.2996

Linshaw, Andrew R.¹; Schwarz, Gerald W.²; Song, Bailin ³

¹ Department of Mathematics University of Denver 2280 S. Vine St. Denver, CO 80208 (USA)
² Department of Mathematics Brandeis University 415 South Street Waltham, MA 02453 (USA)
³ School of Mathematical Sciences University of Science and Technology of China No. 96 Jinzhai Road Hefei, Anhui Province, 230026 (P. R. China)

Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2571-2599.

Abstract
Résumé

Let $G$ be a complex reductive group and $V$ a $G$ -module. Then the $m$ th jet scheme $G_{m}$ acts on the $m$ th jet scheme $V_{m}$ for all $m \geq 0$ . We are interested in the invariant ring $𝒪 {(V_{m})}^{G_{m}}$ and whether the map $p_{m}^{*} : 𝒪 ({(V / / G)}_{m}) \to 𝒪 {(V_{m})}^{G_{m}}$ induced by the categorical quotient map $p : V \to V / / G$ is an isomorphism, surjective, or neither. Using Luna’s slice theorem, we give criteria for $p_{m}^{*}$ to be an isomorphism for all $m$ , and we prove this when $G = {SL}_{n}$ , ${GL}_{n}$ , ${SO}_{n}$ , or ${Sp}_{2 n}$ and $V$ is a sum of copies of the standard module and its dual, such that $V / / G$ is smooth or a complete intersection. We classify all representations of $ℂ^{*}$ for which $p_{\infty}^{*}$ is surjective or an isomorphism. Finally, we give examples where $p_{m}^{*}$ is surjective for $m = \infty$ but not for finite $m$ , and where it is surjective but not injective.

Soient $G$ un groupe réductif complexe et $V$ un $G$ -module. Alors $G_{m}$ , le schéma des jets d’ordre $m$ de $G$ , opère dans $V_{m}$ , le schéma des jets d’ordre $m$ de $V$ , pour tout $m \geq 0$ . Nous nous intéressons à l’anneau des invariants $𝒪 {(V_{m})}^{G_{m}}$ et au morphisme $p_{m}^{*} : 𝒪 ({(V / / G)}_{m}) \to 𝒪 {(V_{m})}^{G_{m}}$ induit par le morphisme du quotient catégorique $p : V \to V / / G$ : ce morphisme est-il un isomorphisme, surjectif, ou non ? En utilisant le théorème du slice de Luna, nous obtenons des critères pour que $p_{m}^{*}$ soit un isomorphisme pour tout $m$ . Nous montrons que c’est bien le cas lorsque $G = {SL}_{n}$ , ${GL}_{n}$ , ${SO}_{n}$ , ou ${Sp}_{2 n}$ et $V$ est un somme directe de copies du module standard et de son dual, pourvu que $V / / G$ soit lisse ou une intersection complète. Nous classifions toutes les représentations de $ℂ^{*}$ telles que $p_{\infty}^{*}$ soit surjectif ou un isomorphisme. Enfin, nous donnons des exemples où $p_{m}^{*}$ est surjectif pour $m = \infty$ mais non surjectif pour $m$ fini, et d’autres exemples où $p_{m}^{*}$ est surjectif mais non injectif.

Zbl: 1342.13009

DOI: 10.5802/aif.2996

Classification: 13A50, 14L24, 14L30
Keywords: jet schemes, classical invariant theory

Author's affiliations:

Linshaw, Andrew R. ¹; Schwarz, Gerald W. ²; Song, Bailin ³

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     title = {Jet schemes and invariant theory},
     journal = {Annales de l'Institut Fourier},
     pages = {2571--2599},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {6},
     year = {2015},
     doi = {10.5802/aif.2996},
     zbl = {1342.13009},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2996/}
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Linshaw, Andrew R.; Schwarz, Gerald W.; Song, Bailin. Jet schemes and invariant theory. Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2571-2599. doi : 10.5802/aif.2996. https://aif.centre-mersenne.org/articles/10.5802/aif.2996/

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